Question

the length, width, and height of a rectangular prism can be represented by the expressions ((x + 3)), ((x + 7)) and ((x - 1)). write an expression to represent the surface area of the prism. ((sa = 2lw + 2lh + 2wh))

Explanation:

Step1: Identify length, width, height

Let \( l = x + 3 \), \( w = x + 7 \), \( h = x - 1 \). The surface area formula is \( SA = 2lw + 2lh + 2wh \).

Step2: Calculate \( 2lw \)

\[

$$\begin{align*} 2lw &= 2(x + 3)(x + 7)\\ &= 2(x^2 + 7x + 3x + 21)\\ &= 2(x^2 + 10x + 21)\\ &= 2x^2 + 20x + 42 \end{align*}$$

\]

Step3: Calculate \( 2lh \)

\[

$$\begin{align*} 2lh &= 2(x + 3)(x - 1)\\ &= 2(x^2 - x + 3x - 3)\\ &= 2(x^2 + 2x - 3)\\ &= 2x^2 + 4x - 6 \end{align*}$$

\]

Step4: Calculate \( 2wh \)

\[

$$\begin{align*} 2wh &= 2(x + 7)(x - 1)\\ &= 2(x^2 - x + 7x - 7)\\ &= 2(x^2 + 6x - 7)\\ &= 2x^2 + 12x - 14 \end{align*}$$

\]

Step5: Sum the three terms

\[

$$\begin{align*} SA &= (2x^2 + 20x + 42) + (2x^2 + 4x - 6) + (2x^2 + 12x - 14)\\ &= (2x^2 + 2x^2 + 2x^2) + (20x + 4x + 12x) + (42 - 6 - 14)\\ &= 6x^2 + 36x + 22 \end{align*}$$

\]

Answer:

The expression for the surface area is \( 6x^2 + 36x + 22 \)